multivariatemethods - sabancı Üniversitesipeople.sabanciuniv.edu/berrin/cs512/lectures/... · 4...

Multivariate Methods

Slides from Machine Learning by Ethem Alpaydin Expanded by some slides from Gutierrez-‐‑Osuna

Overview

n  We learned how to use the Bayesian approach for classification if we had the probability distribution of the underlying classes (p(x|Ci)).

n  Now going to look into how to estimate those densities from given samples.

Expectations

Approximate Expecta.on (discrete and con.nuous)

The average value of a function f(x) under a probability distribution p(x) is called the expectation of f(x). Average is weighted by the relative probabilities of different values of x.

Now we are going to look at concepts: variance and co-variance , of 1 or more random variables, using the concept of expectation.

Variance and Covariance

The variance of f(x) provides a measure for how much f(x) varies around its mean E[f(x)]. Given a set of N points {xi} in the1D-space, the variance of the corresponding random variable x is var[x] = E[(x-µ)2] where µ=E[x]. You can estimate the expected value as

var(x) = E[(x-µ)2] ≈ 1/N Σ (xi-µ)2 x

i Remember the definition of expectation:

The variance of x provides a measure for how much x varies around its mean µ=E[x]. var(x) = E[(x-µ)2] Co-variance of two random variables x and y measures the extent to which they vary together.

Co-variance of two random variables x and y measures the extent to which they vary together.

: two random variables x, y : two vector random variables x, y – covariance is a matrix

Multivariate Normal Distribution

The Gaussian Distribution

Expectations

For normally distributed x:

n  Assume we have a d-dimensional input (e.g. 2D), x.

n  We will see how can we characterize p(x), assuming x is normally distributed.

¨  For 1-dimension it was the mean (µ) and variance (σ2)

n  Mean=E[x]

n  Variance=E[(x - µ)2]

¨  For d-dimensions, we need n  the d-dimensional mean vector n  dxd dimensional covariance matrix

¨  If x ~ Nd(µ,Σ) then each dimension of x is univariate normal n  Converse is not true

Normal Distribution & Multivariate Normal Distribution

n  For a single variable, the normal density function is:

n  For variables in higher dimensions, this generalizes to:

n  where the mean µ is now a d-dimensional vector, Σ is a d x d covariance matrix

|Σ| is the determinant of Σ:

Multivariate Parameters: Mean, Covariance

( ) TTT

EE µµxxµxµxx −=−−=

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=≡Σ ][]))([(Cov

σσσ

[ ] [ ]TdE µµ ,...,:Mean 1== µx

( ) )])([(,Cov jjiijiij xxExx µµσ −−≡≡

Matlab code

n  close all;

n  rand('twister', 1987) % seed

%Define the parameters of two 2d-Normal distribution

n  mu1 = [5 -5];

n  mu2 = [0 0];

n  sigma1 = [2 0; 0 2];

n  sigma2 = [5 5; 5 5];

n  N=500; %Number of samples we want to generate from this distribution

n  samp1 = mvnrnd(mu1,sigma1, N);

n  samp2 = mvnrnd(mu2, sigma2, N);

n  figure; clf;

n  plot(samp1(:,1), samp1(:,2),'.', 'MarkerEdgeColor', 'b');

n  hold on;

n  plot(samp2(:,1), samp2(:,2),'*', 'MarkerEdgeColor', 'r');

n  axis([-20 20 -20 20]); legend('d1', 'd2'); 13

mu1 = [5 -‐‑5]; mu2 = [0 0]; sigma1 = [2 0; 0 2]; sigma2 = [5 5; 5 5];

mu1 = [5 -‐‑5]; mu2 = [0 0]; sigma1 = [2 0; 0 2]; sigma2 = [5 2; 2 5];

Matlab sample cont.

n  % Lets compute the mean and covariance as if we are given this data

n  sampmu1 = sum(samp1)/N;

n  sampmu2 = sum(samp2)/N;

n  sampcov1 = zeros(2,2);

n  sampcov2 = zeros(2,2);

n  for i =1:N

n  sampcov1 = sampcov1 + (samp1(i,:)-sampmu1)' * (samp1(i,:)-sampmu1);

n  sampcov2 = sampcov2 + (samp2(i,:)-sampmu2)' * (samp2(i,:)-sampmu2);

n  End

n  sampcov1 = sampcov1 /N;

n  sampcov2 = sampcov2 /N;

n  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n  % Lets compute the mean and covariance as if we are given this data USING MATRIX OPERATIONS n  % Notice that in samp1, samples are given in ROWS – but for this multiplication, columns * rows is req. n  sampcov1 = (samp1'*samp1)/N - sampmu1'*sampmu1; n  %Or simply

n  mu=mean(samp1);

n  cov=cov(samp1); 16

n  Variance: How much X varies around the expected value

n  Covariance is the measure the strength of the linear relationship between two random variables ¨  covariance becomes more positive

for each pair of values which differ from their mean in the same direction

¨  covariance becomes more negative with each pair of values which differ from their mean in opposite directions.

¨  if two variables are independent, then their covariance/correlation is zero (converse is not true).

ijijji

jjiijiij

µµσ

−−≡≡

,Corr :nCorrelatio

)])([(,Cov:Covariance

n  Correlation is a dimensionless measure of linear dependence. ¨  range between –1 and +1

How to characterize differences between these distributions

Covariance Matrices

Contours of constant probability density for a 2D Gaussian distributions with a) general covariance matrix b) diagonal covariance matrix (covariance of x1,x2 is 0) c) Σ proportional to the identity matrix (covariances are 0, variances of each dimension are the same)

Shape and orientation of the hyper-ellipsoid centered at µ is defined by Σ

Properties of Σ

n  A small value of |Σ| (determinant of the covariance matrix) indicates that samples are close to µ

n  Small |Σ| may also indicate that there is a high correlation between variables

n  If some of the variables are linearly dependent, or

if the variance of one variable is 0, then Σ is singular and |Σ| is 0.

n  Dimensionality should be reduced to get a positive definite matrix

From the equation for the normal density, it is apparent that points which have the same density must have the same constant term: Mahalanobis distance measures the distance from x to μ in terms of ∑

⎥⎦⎤

⎢⎣⎡ −−−= − )()(21exp

)2(1)( 1

2/12/µxΣµx

πMahalanobis Distance

Points that are the same distance from µ

n  The

• The ellipse consists of points that are equi-distant to the center w.r.t. Mahalanobis distance. • The circle consists of points that are equi-distant to the center w.r.t. The Euclidian distance.

Why Mahalanobis Distance

It takes into account the covariance of the data. •  Point P is at closer (Euclidean) to the mean for the orange class, but using the Mahalanobis distance, it is found to be closer to 'apple‘ class.

Positive Semi-Definite-Advanced

n  The covariance matrix Σ is symmetric and positive semi-definite

n  An n × n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries.

n  An n × n real symmetric matrix M is positive semi-definite

if zTMz >= 0 for all non-zero vectors z with real entries. n  Comes from the requirement that the variance of each dimension is >= 0

and that the matrix is symmetric.

n  When you randomly generate a covariance matrix, it may violate this rule ¨  Test to see if all the eigenvalues are >= 0 ¨  Higham 2002 – how to find nearest valid covariance matrix ¨  Set negative eigenvalues to small positive values

Parameter Estimation

Covered only ML estimator

n  You have some samples coming from an unknown distribution and you want to characterize that distribution; i.e. find the necessary parameters.

n  For instance, assuming you are given the samples in Slides 14 or 15, and that you assume that they are normally distributed, you will try to find the parameters µ and Σ of the corresponding normal distribution.

n  We have referred to the sample mean (mean of the samples, m) and sample variance S before, but why use those instead of µ and Σ?

n  Given an i.i.d data set X, sampled from a normal distribution with parameters µ and Σ, we would like to determine these parameters, given the samples.

n  Once we have those, we can estimate p(x) for any given x (little x), given the known distribution.

n  Two approaches in parameter estimation: ¨  Maximum Likelihood approach ¨  Bayesian approach

Gaussian Parameter Estimation

Likelihood func.on

Assuming iid data

Reminder

n  In order to maximize or minimize a function f(x) w.r.to x, we compute the derivative df(x)/dx and set to 0, since a necessary condition for extrema is that the derivative is 0.

n  Commonly used derivative rules are given on the right.

Derivation – general case-ADVANCED

Maximum (Log) Likelihood for Multivarate Case

Maximum (Log) Likelihood for a 1D-Gaussian

In other words, maximum likelihood estimates of mean and variance are the same as sample mean and variance.

Sample Mean and Variance for Multivariate Case

( )( )

−−=

:matrix n Correlatio

:matrix Covariance

,...,1,:mean Sample

Where N is the number of data points xt:

ML estimates for the mean and variance in Bernouilli/Multinomial distributions

Not covered in class – but you should be able to do as a take home etc. Same idea as before, starting from the likelihood, you find the value that maximizes the likelihood.

Examples: Bernoulli/Multinomial

n  Bernouilli: Binary random variable x may take one of the two values: ¨  success/failure, 0/1 with probabilities: ¨  P(x=1)= po

¨  P(x=0)=1-po ¨  Unifying the above, we get: P (x) = po

x (1 – po ) (1 – x)

n  Given a sample set X={x1,x2,…}, we can estimate p using the ML

estimate by maximizing the log-likelihood of the sample set:

Log likelihood: log P(X|po) = log ∏t po

xt (1 – po ) (1 – xt)

Examples: Bernoulli

L=P(X|po) = log ∏t po

xt (1 – po ) (1 – xt) xt in {0,1}

Solving for the necessary condition for extrema, (we must have dL/dp = 0)

MLE: po = ∑

t xt / N

ratio of the number of occurences of the event to the number of experiments

Examples: Multinomial

n  Multinomial: K>2 states, xi in {0,1}

n  Instead of two states, we now have K mutually exclusive and exhaustive events, with probability of occurence of

pi where Σpi = 1.

n  Ex. A dice with 6 outcomes.

P (x1,x2,...,xK) = ∏i pi

xi where xi is 1 if the outcome is state i 0 otherwise

L(p1,p2,...,pK|X) = log ∏

MLE: pi = ∑

t / N Ratio of experiments with outcome of state i (e.g. 60 dice throws, 15 of them were 6 p6 = 15/60

Discrete Features

n  Binary features:

If xj are independent (Naive Bayes’ assumption)

( ) ( ) ( )( ) ( )[ ] ( )i

jijjijj

CPpxpxCPCpg

log1 log 1 log log| log

+−−+=

Estimated parameters

( )ijij Cxpp |1=≡

( ) ( )( )∏=

−−=d

jj ppCxp1

∑∑

ri=1 if xt in Ci

0 otherwise

Discrete Features

n  Multinomial (1-of-nj) features: xj ∈ {v1, v2,..., vnj}

if xj are independent

( ) ( )ikjijkijk CvxpCzpp ||1 ===≡

( ) ( )

∑∑∑ ∑

∏∏

iijkj k jki

log log

Parametric Classification

We will use the Bayesian decision criteria applied to normally distributed classes, whose parameters are either known or estimated from the sample.

Parametric Classification

n  If p (x | Ci ) ~ N ( μi , ∑i )

n  Discriminant functions are:

( )( )

( ) ( )⎥⎦⎤

⎢⎣

⎡ −−−= −ii

idiCp µxµxx 1

2/12/ Σ21exp

( )( ) ( )

( ) ( ) ( )iiiT

CPdCPCp

logµΣµ21Σ log

212log

log| log )|(log

1 +−−−−−=

− xx

Estimation of Parameters

( ) ( ) ( ) ( )iiiT

iii CPg ˆ log21 log

21 1 +−−−−= − mxmxx SS

If we estimate the unknown parameters from the sample, the discriminant function becomes:

Case 1) Different Si (each class has a separate variance)

( ) ( ) ( )

( )iiiiTii

iiiTiii

log log21

log221

+−−=

++−−−=

−−−

mmmxxxx

if we group the terms, we see that there are second order terms, which means the discriminant is quadratic.

likelihoods

posterior for C1

discriminant: P (C1|x ) = 0.5

Two bi-variate normals, with completely different covariance matrix, showing a hyper-quadratic decision boundary.

Hyperbola: A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the

axis of the cone. Definition from Wikipedia.

Typical single-variable normal distributions showing a disconnected decision region R2

Notation: Ethem book

( )( )∑

∑∑∑

−−=

Using the notation in Ethem book, the sample mean and sample covariance… can be estimated as follows:

where is 1 if the tth sample belongs to class i

n  If d (dimension) is large with respect to N (number of samples), we may have a problem with this approach: ¨  | Σ | may be zero, thus Σ will be singular (inverse does not exist) ¨  | Σ | may be non-zero, but very small, instability would result

n  Small changes in Σ would cause large changes in Σ-1

n  Solutions: ¨  Reduce the dimensionality

n  Feature selection n  Feature extraction: PCA

¨  Pool the data and estimate a common covariance matrix for all classes

Σ = Σi P(Ci) * Σi

Case 2) Common Covariance Matrix S=Si

n  Shared common sample covariance S ¨  An arbitrary covariance matrix – but shared between the classes

n  We had this full discriminant function:

which now reduces to:

which is a linear discriminant

( ) ( ) ( ) ( )iiT

ii CPg ˆ log21 1 +−−−= − mxmxx S

( )iiTiiii

ˆ log21

+−==

−− mmmw

( ) ( ) ( ) ( )iiiT

iii CPg ˆ log21 log

21 1 +−−−−= − mxmxx SS

n  Linear discriminant: ¨  Decision boundaries are hyper-planes ¨  Convex decision regions:

n  All points between two arbitrary points chosen from one decision region belongs to the same decision region

n  If we also assume equal class priors, the classifier becomes a minimum Mahalanobis classifier

Case 2) Common Covariance Matrix S

Linear discriminant

Unequal priors shift the decision boundary towards the less likely class, as before.

Case 3) Common Covariance Matrix S which is Diagonal

n  In the previous case, we had a common, general covariance matrix, resulting in these discriminant functions:

n  When xj (j = 1,..d) are independent, ∑ is diagonal

n  Classification is done based on weighted Euclidean distance (in sj units) to the nearest mean.

Naive Bayes classifier where p(xj|Ci) are univariate Gaussian

p (x|Ci) = ∏j p (xj |Ci) (Naive Bayes’ assumption)

( ) ( )id

i CPsmx

g ˆ log21

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= ∑

( ) ( ) ( ) ( )iiT

ii CPg ˆ log21 1 +−−−= − mxmxx S

Case 3) Common Covariance Matrix S which is Diagonal

variances may be different

Case 4) Common Covariance Matrix S which is Diagonal + equal variances

n  We had this before (S which is diagonal):

n  If the priors are also equal, we have the Nearest Mean classifier: Classify based on Euclidean distance to the nearest mean!

n  Each mean can be considered a prototype or template and this is template matching

( ) ( )

( ) ( )id

ˆ log21

ˆ log2

+−−=

( ) ( )id

i CPsmx

g ˆ log21

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= ∑

Case 4) Common Covariance Matrix S which is Diagonal + equal variances

A second example where the priors have been changed: The decision boundary has shifted away from the more likely class, although it is still orthogonal to the line joining the 2 means.

Case 5: Si=σi2I

Model Selection

n  As we increase complexity (less restricted S), bias decreases and variance increases

n  Assume simple models (allow some bias) to control variance (regularization)

Assumption Covariance matrix No of parameters

Shared, Hyperspheric Si=S=s2I 1

Shared, Axis-aligned Si=S, with sij=0 d

Shared, Hyperellipsoidal Si=S d(d+1)/2

Different, Hyperellipsoidal Si K d(d+1)/2

Estimation of Missing Values

n  What to do if certain instances have missing attributes? 1.  Ignore those instances: not a good idea if the sample is small 2.  Use ‘missing’ as an attribute: may give information 3.  Imputation: Fill in the missing value

n  Mean imputation: Use the most likely value (e.g., mean) n  Imputation by regression: Predict based on other attributes

n  Another important problem is sensitivity due to outliers

Tuning Complexity

n  When we use Euclidian distance to measure similarity, we are assuming that all variables have the same variance and that they are independent ¨  E.g. Two variables age and yearly income

n  When these assumptions don’t hold, ¨  Normalization may be used (use PCA, whitening, make each dimension

zero mean and unit variance…) to use Euclidian distance ¨  We may still want to use simpler models in order to estimate the related

parameters more accurately

Conclusions

n  The Bayes classifier for normally distributed classes is a quadratic classifier

n  The Bayes classifier for normally distributed classes with equal covariance matrices is a linear classifier

n  The minimum Mahalanobis distance is Bayes-optimal for ¨  Normally distributed classes, having ¨  Equal covariance matrices and ¨  Equal priors

n  The minimum Euclidian distance is Bayes-optimal for ¨  Normally distributed classes -and- ¨  Equal covariance matrices proportional to the identity matrix–and- ¨  Equal priors

n  Both Euclidian and Mahalanobis distance classifiers are linear classifiers

PRTOOLS n  load nutsbolts; //an existing Prtools dataset with 4 classes n  w1=ldc(z); //linear Bayes Normal classifier n  w2=qdc(z); //quadratic Bayes Normal classifier n  figure(1); scatterd(z); hold on; plotc(w1); n  figure(2); scatterd(z); hold on; plotc(w2);

n  >> A = gendath([50 50]); //Generate data with 2 classes x 50 samples each n  >> [C,D] = gendat(A,[20 20]); //Split 20 as C=train; rest becomes D=test n  >> W1 = ldc(C); //linear n  >> W2 = qdc(C); //quadratic n  >> figure(5); scatterd(C); hold on; plotm(W2);

n  scatterd(A); //scatter plot n  plotc({W1,W2});

error on test: 9.5%

error on test: 11.5%

with N=20 points

Eigenvalue Decomposition of the Covariance Matrix

Skip Until Dimensionality Reduction Topic

Eigenvector and Eigenvalue

¨  Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation Ax = λ x for some scalar λ.

n  In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.

n  (A- λI)x=0 => det(A- λI) must be 0. ¨  Gives the characteristic polynomial whore roots are the eigenvalues of A

The covariance matrix is symmetrical and it can always be diagonalized as:

TΦΛΦ=Σ

where is the column matrix consisting of the eigenvectors of Σ, ΦT=Φ-‐‑1 and Λ is the diagonal matrix whose elements are the eigenvalues of Σ.

],...,,[ 21 lυυυ=Φ

Eigenvalues of the covariance matrix

Contours of equal Mahalanobis distance:

( ) cxxd iT

im =−Σ−= − 2/11 )()( µµ

Define x’= ΦTx. The coordinates of x’ are equal to νkTx, k=1,2,…,l

that is, the projections of x onto the eigenvectors.

21 )()( cxx iTT

i =−ΦΦΛ− − µµ

211 )(...)( cxx

illi =ʹ′−ʹ′

++ʹ′−ʹ′

Thus all points having the same Mahalabonis distance from a specific point are located on an ellipse with center of mass at µi, and the principle axes are aligned with the corresponding eigenvectors and have lengths ckλ2

n  Eigenvalue decomposition of the covariance matrix is very useful: ¨  PCA ¨  Decorrelate data (Whitening transform) ¨  …

Matrix Transpose

n  Given the matrix A, the transpose of A is the , denoted AT, whose columns are formed from the corresponding rows of A.

n  Some properties of transpose 1.  (AT)T = A

2.  (A + B)T = AT + BT

3.  (rA)T = rAT where r is any scalar.

4.  (AB)T = BTAT

5.  ATB=BTA where A and B are vectors

6.  …

Matrix Derivation

n  If y = xTAx where A is a square matrix ¨  ∂y/∂x = Ax + AT x

n  If y = xTAx and A is symmetric ¨  ∂y/∂x = 2Ax.

n  If y = xTx ¨  ∂y/∂x = 2x.

n  If x ~ Nd(µ,Σ) then each dimension of x is univariate normal ¨  Converse is not true – counterexample?

n  The projection of a d-dimensional normal distribution onto the vector w is univariate normal

wTx ~ N (wTµ, wTΣw)

n  More generally, when W is a dxk matrix with k < d, then the k-dimensional Wx is k-variate normal with:

WTx ~ Nk(WTµ, WTΣW)

Univariate normal – transformed x

n  E[wTx] = wT E[x] = wT µ

n  Var[wTx] = E[ (wTx - wT µ) (wTx - wT µ)Τ]

= E[ (wTx - wT µ)2] since (wTx - wT µ) is a scalar = E[ (wTx - wT µ) (wTx - wT µ) ]

= Ε[ wT(x-µ) (x-µ)Τ w ] based on slide 27.rule 5 = wTΣ w move out and use defin. of Σ

Whitening Transform

n  We can decorrelate variables and obtain Σ=I by using a whitening transform Aw=Λ-1/2ΦT where ¨  Λ is the diagonal matrix of the eigenvalues of the original distribution

and ¨  Φ is the matrix composed of eigenvectors as its columns

multivariatemethods - sabancı Üniversitesipeople.sabanciuniv.edu/berrin/cs512/lectures/... · 4...

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